On the analyticity of solutions to non-linear elliptic partial differential equations

We give an easy proof of the fact that $C^\infty$ solutions to non-linear elliptic equations of second order $$\phi(x, u, D u, D^2 u)=0$$ are analytic. Following ideas of Kato, the proof uses an inductive estimate for suitable weighted derivatives. We then conclude the proof using Cauchy's method of majorants}.